Filtration on a group

A filtration on a group $G$ is a function $\omega :G\to \mathbb{R}\cup \{\infty \}$ satisfyingIt is separated when $\omega (x)=\infty$ only for $x=1$.

For a prime number $p$, a p-valuation is a separated filtration on a group such that every $x\ne 1$ satisfiesA group equipped with one is called a p-valued group.

An ordered basis $(g_1,\dots ,g_d)$ of a complete finite-rank p-valued group gives each element a unique convergent expression $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i\in {\mathbb{Z}}_p$, and its valuation is the minimum of $\omega (g_i)+v_p({\lambda }_i)$ over the nonzero coordinates.

A p-valued group is p-saturated when it is complete and every $x$ withhas a pth root in the group.

For $G_{\lambda }=\{g:\omega (g)\ge \lambda \}$ and $G_{\lambda +}=\{g:\omega (g)>\lambda \}$, the associated graded groupis a graded Lie algebra with bracket induced by the group commutator. For a p-valuation, $t{\mathrm{gr}}_{\lambda }(g)={\mathrm{gr}}_{\lambda +1}(g^p)$ makes it a graded ${\mathbb{F}}_p[t]$-Lie algebra.